On disjoint configurations in infinite graphs

For a graph A and a positive integer n , let nA denote the union of n disjoint copies of A ; similarly, the union of ℵ 0 disjoint copies of A is referred to as ℵ 0 A . It is shown that there exist (connected) graphs A and G such that nA is a minor of G for all n ϵℕ, but ℵ 0 A is not a minor of G . This supplements previous examples showing that analogous statements are true if, instead of minors, isomorphic embeddings or topological minors are considered. The construction of A and G is based on the fact that there exist (infinite) graphs G 1 , G 2 ,… such that G i is not a minor of G j for all i  ≠  j . In contrast to previous examples concerning isomorphic embeddings and topological minors, the graphs A and G presented here are not locally finite. The following conjecture is suggested: for each locally finite connected graph A and each graph G , if nA is a minor of G for all n  ϵ ℕ, then ℵ 0 A is a minor of G , too. If true, this would be a far‐reaching generalization of a classical result of R. Halin on families of disjoint one‐way infinite paths in graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 39: 222–229, 2002; DOI 10.1002/jgt.10016